### Upcoming Talks

#### Habilitationsvortrag (Lehrprobe)

Title: The Tarski-Seidenberg Principle
Speaker: Dr. Christopher Frei (Univ. of Manchester)
Date: Freitag, 15. 12. 2017, 14:00 Uhr
Room: SR Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24, 2.Stock
Abstract:

#### Doktoratskolleg Discrete Mathematics

Title:
Speaker: Discrete Mathematics Day 2017 ()
Date: Thursday, 14.12.2017, 10:30-16:40
Room: Hörsaal BE01, Steyregasse 30, EG
Abstract:

10:30 opening
10:40-11:30 main talk 1: Christopher Frei (Manchester)
11:30-10:40 Math.Video
10:45-12:15 PhD talk 1: JunSeok Oh (KFU Graz)
12:15-12:25 Math.Video
12:25-13:30 Lunch buffet
13:45-14:15 PhD talk 2: Shu-Qin Zhang (MU Leoben)
14:15-14:25 Math.Video
14:30-15:00 PhD talk 3: Irene de Parada (TU Graz)
15:00-15:10 Math.Video
15:10-15:40 Coffee break
15:40-16:30 main talk 2: Silke Rolles (TU München)
16:30-16:40 Math.Video

A more detailed programme will follow.

#### Strukturtheorie-Seminar

Title: Reinforced random walk
Speaker: Michael Kalab (TU Graz)
Date: Donnerstag, 7.12.2017, 11 Uhr c.t.
Room: Seminarraum AE02, Steyrergasse 30, Erdgeschoss
Abstract:

In this master-seminar, linearly reinforced random walks are explained and some results are presented.

#### FWF START Seminar (Mini-Colloquium)

Title: Inhomogeneous Diophantine Approximation with Restricted Denominators
Speaker: Agamemnon Zafeiropoulos (TU Graz)
Date: 4.12.2017, 16:00
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

We formulate and prove a Khintchine-type law for inhomogeneous Diophantine approximation. The denominators form a lacunary sequence of integers, while the size of the set of well-approximable numbers is given with respect to a probability measure with Fourier coefficients of a prescribed logarithmic decay rate.

(Remark: Agamemnon Zafeiropoulos is a new member of the Institute of Analysis and Number Theory, who started here as a Postdoc researcher in November 2017.)

#### FWF START Seminar (Mini-Colloquium)

Title: Joint universality for dependent L-functions
Speaker: Lukasz Pankowski (Adam Mickiewicz University Poznan)
Date: 4.12.2017, 15:15
Room: Seminarraum Analysis-Zahlentheorie, Kopernikusgasse 24, 2.OG
Abstract:

We prove that, for arbitrary Dirichlet $L$-functions $L(s;\chi_1),\ldots, L(s;\chi_n)$ (including the case when $\chi_j$ is equivalent to $\chi_l$ for $j\ne k$), suitable shifts of type $L(s+i\alpha_jt^{a_j}\log^{b_j}t;\chi_j)$ can simultaneously approximate any given analytic functions on a simply connected compact subset of the right open half of the critical strip, provided the pairs $(a_j,b_j)$ are distinct and satisfy certain conditions. Moreover, we consider a discrete analogue of this problem where $t$ runs over the set of positive integers.

#### Vortrag im Seminar für Kombinatorik und Optimierung

Title: Tangle Crossing Numbers and theTanglegram Kuratowski Theorem
Speaker: \'Eva Czabarka (University of South Carolina)
Date: Freitag 1. Dezember, 15:00
Room: Hörsaal BE01, Steyrergasse 30, Erdgeschoss
Abstract:

A tanglegram of size $n$ is a triplet $(L,R,M)$ where $L$ and $R$ are rooted binary trees with $n$ leaves each, and $M$ is a perfect matching between the two sets of leaves. Two tanglegrams $(L_1,R_1,M_1)$ and $(L_2,R_2,M_2)$ are the same if there is a pair of tree-isomorphisms $(\phi,\psi)$ mapping $L_1$ to $L_2$ and $R_1$ to $R_2$ such that matched pairs of leaves get paired to matched pairs of leaves. Tanglegrams are used in phylogenetics, where for example they can represent the phylogenetic trees of parasites and hosts, where the matching gives which parasite infects which host.

A tanglegram layout (i.e. the way tanglegrams are usually drawn) is as follows: draw the two rooted binary trees in the plane with straight lines and without crossing edges such that the leaves of $L$ are on the line $x=0$ and $L$ is drawn in the semi-plane $x\le 0$, the leaves of $R$ are drawn on the line $x=1$ and $R$ is drawn in the semi-plane $x\ge 1$, and the edges of the matching are drawn with straight line. The crossing number of a layout is the number of unordered pairs of matching edges that cross and the tangle crossing number of a tanglegram is the minimum crossing number over all of its layouts. The tangle crossing number is related to a number of biologically important quantities, e.g. the number of times parasites switched hosts. I will present some results about the tangle crossing number, including a Kuratowski type theorem.

#### Vortrag im Seminar für Kombinatorik und Optimierung

Title: The Partition Adjacency Matrix realization problem
Speaker: L\'aszl\'o Sz\'ekely (University of South Carolina)
Date: Freitag 1. Dezember, 14:00 Kaffeepause 13:30
Room: Hörsaal BE01, Steyrergasse 30, Erdgeschoss
Abstract:

On Facebook, people with high number of connections tend to be connected more likely than randomness would suggest, while in biological networks vertices with high number of connections tend to be connected less likely than randomness would suggest. In terms of network science, the first network is assortative, while the second is disassortative.

Degrees (number of connections) do not tell if a network is assortative or disassortative. The Joint Degree Matrix (JDM) of a network (graph) counts number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem asks whether a graph exists with prescribed number of connections (degree) at the vertices, and with prescribed number of edges between the sets of degree $i$ and degree $j$ vertices, for any $i,j$. The JDM realization problem is well understood. The usual measure for assortativity, the assortativity coefficient, the Pearson correlation coefficient of degree between pairs of linked nodes, is computable from the JDM.

A further generalization of the JDM is the following.
Given a set $W$ and numbers $d(w)$ associated with $w\in W$, and a $W_i:i\in I$ partition of $W$, with numbers $c(W_i,W_j)$ associated with unordered pairs of partition classes, the Partition Adjacency Matrix (PAM) realization problem asks whether there is a simple graph $G$ on the vertex set $W$ with degrees $d_G(w)=d(w)$ for $w\in W$, with exactly $c(W_i,W_j)$ edges with endpoints in $W_i$ and $W_j$; and the PAM construction problem asks for such a graph, if they exist. (These problems are conjectured to be NP-hard.) The bipartite version of these problems are more restricted: $I=I_1\cup I_2$ and $c(W_i,W_j)=0$ whenever $i,j\in I_1$ or $i,j\in I_2$.

We provide algebraic Monte-Carlo algorithms for the bipartite Partition Adjacency Matrix realization and construction problems, which run in polynomial time, say, when $|I|$ is fixed. When the algorithms provide a positive answer, they are always correct, and when the truth is positive, the algorithms fail to report it with small probability.

#### Zahlentheoretisches KolloquiumACHTUNG - Die Beginnzeit des Vortrages hat sich geändert!

Title: Metric discrepancy results for geometric progressions with small ratios 3/2, 4/3, etc.
Speaker: Prof. Dr. Katusi Fukuyama (Kobe University, Japan)
Date: Dienstag, 28. 11. 2017, 12:00 Uhr
Room: Seminarraum Analysis-Zahlentheorie (NT02008), Kopernikusgasse 24/II
Abstract:

#### Sturkturtheorie-Seminar

Title: The connective constant
Speaker: Christian Lindorfer (TU Graz)
Date: Donnerstag, 23.11.2017, 11 Uhr c.t.
Room: Seminarraum AE02, Steyrergasse 30, Erdgeschoss
Abstract:

In this master seminar, self-avoiding walks on infinite graphs are discussed,
with focus on Cayley graphs and quasi-transitive graphs.
The connective constant is the exponential growth rate of the number of self-avoiding walks of length n. Its computation for lattices is a difficult problem coming from Statistical Physics. In the talk, some basic properties, recent results, and computations are presented.